The problem is that if the performance gap is imaginary while the pay gap is real, then someone could take advantage of the situation for enormous profit. If other employers falsely believe that some much more expensive hammers will be better at driving nails, that's great news for anyone who recognizes that the cheap hammers perform just as well--when 99% of my competitors are wildly deluded, there is the potential for profit.
Thus, we have two possibilities: either nobody in the world is both aware of the potential for arbitrage and unethical enough to exploit the situation, or the wage gap is smaller than these estimates suggest.
You're assuming that public image of said employers is unimportant, and that all employers are okay with being immoral. There's something to be said about companies with happy employees that don't feel like their employers are taking advantage of them.
There's nothing stopping employers from making the practical decision to employ underpriced women yet couch it in the rhetoric of feminism and diversity ("Diversity is important to us, and that's why we want to employ as many qualified women as possible.") This gets them a PR and economic benefit.
In fact, many new-economy juggernauts like Microsoft, Google, and Facebook do precisely this. Their employee marketing aggressively recruits women. Are they doing this out of the goodness of their hearts, or because they see an economic opportunity? Ultimately it doesn't really matter - the effect is that more women get employed in positions of power. But it's quite possible that the economic arbitrage that people are pointing out is happening in practice, right now, you just don't hear about it because most arbitragers are not dumb enough to let you know what they're doing. And the effect, over a long time scale, is that businesses run by bigots are replaced by businesses with a more accurate perception of personal productivity, which leads to a more equitable world for everyone.
Thus, we have two possibilities: either nobody in the world is both aware of the potential for arbitrage and unethical enough to exploit the situation, or the wage gap is smaller than these estimates suggest.